module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.MetricSpace.Thickening | {
"line": 63,
"column": 52
} | {
"line": 67,
"column": 54
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nE : Set α\nx : α\nh : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\n⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ thickening δ E",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
... | [] | by
obtain ⟨ε, ⟨ε_pos, ε_lt⟩⟩ := exists_real_pos_lt_infEDist_of_notMem_closure h
filter_upwards [eventually_lt_nhds ε_pos] with δ hδ
simp only [thickening, mem_setOf_eq, not_lt]
exact (ENNReal.ofReal_le_ofReal hδ.le).trans ε_lt.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 364,
"column": 2
} | {
"line": 369,
"column": 36
} | {
"line": 371,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nε : ℝ≥0\nx y : α\nhy : infEDist y s ≤ ↑ε\nδ : ℝ≥0\nhδ : 0 < δ\nx✝ : ediam s + 2 * ↑ε < ∞\nhε : ↑ε < ↑ε + ↑δ\nhx : infEDist x s < ↑ε + ↑δ\nx' : α\nhx' : x' ∈ s\nhxx' : edist x x' < ↑ε + ↑δ\n⊢ edist x y ≤ ediam s + 2 * ↑ε + ↑δ",
"ppTerm": "?m.125",... | [] | calc
edist x y ≤ edist x x' + edist y x' := edist_triangle_right _ _ _
_ ≤ ε + δ + (infEDist y s + ediam s) :=
add_le_add hxx'.le (edist_le_infEDist_add_ediam hx')
_ ≤ ε + δ + (ε + ediam s) := by grw [hy]
_ = _ := by rw [two_mul]; ac_rfl | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 99,
"column": 2
} | {
"line": 110,
"column": 84
} | {
"line": 112,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE✝ : Type u_3\nF : Type u_4\nα : Type u_5\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup E✝\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 E✝\ninst✝² : NormedSpace 𝕜 F\nE : Type u_6\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℚ E\ne : E\n⊢... | [] | have : IsAddTorsionFree E := .of_module_rat E
rcases eq_or_ne e 0 with (rfl | he)
· rw [AddSubgroup.zmultiples_zero_eq_bot]
exact Subsingleton.discreteTopology (α := ↑(⊥ : Subspace ℚ E))
· rw [discreteTopology_iff_isOpen_singleton_zero, isOpen_induced_iff]
refine ⟨Metric.ball 0 ‖e‖, Metric.isOpen_ball, ?_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 99,
"column": 2
} | {
"line": 110,
"column": 84
} | {
"line": 112,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE✝ : Type u_3\nF : Type u_4\nα : Type u_5\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup E✝\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 E✝\ninst✝² : NormedSpace 𝕜 F\nE : Type u_6\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℚ E\ne : E\n⊢... | [] | have : IsAddTorsionFree E := .of_module_rat E
rcases eq_or_ne e 0 with (rfl | he)
· rw [AddSubgroup.zmultiples_zero_eq_bot]
exact Subsingleton.discreteTopology (α := ↑(⊥ : Subspace ℚ E))
· rw [discreteTopology_iff_isOpen_singleton_zero, isOpen_induced_iff]
refine ⟨Metric.ball 0 ‖e‖, Metric.isOpen_ball, ?_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 205,
"column": 7
} | {
"line": 205,
"column": 19
} | {
"line": 205,
"column": 19
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : Nontrivial E\nc : ℝ\nx : E\nhx : x ≠ 0\nr : 𝕜\nhr : c / ‖x‖ < ‖r‖\n⊢ 0 < ‖x‖",
"ppTerm": "?h✝",
"assigned": true,
"usedConstants": [
"AddGroup.t... | [
"case h\n𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : Nontrivial E\nc : ℝ\nx : E\nhx : x ≠ 0\nr : 𝕜\nhr : c / ‖x‖ < ‖r‖\n⊢ x ≠ 0"
] | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 293,
"column": 82
} | {
"line": 294,
"column": 41
} | {
"line": 296,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nx : 𝕜\n⊢ ‖(algebraMap 𝕜 𝕜') x‖ = ‖x‖",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.to... | [] | by
rw [norm_algebraMap, norm_one, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 292,
"column": 2
} | {
"line": 293,
"column": 44
} | {
"line": 294,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm : MeasurableSpace α\nf : α → β\nm' m₀ : MeasurableSpace α\nμ : Measure α\ninst✝ : Zero β\nhm : m ≤ m₀\ns : Set α\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)\nhf : AEStronglyMeasurable f μ\nhf_zero ... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm : MeasurableSpace α\nf : α → β\nm' m₀ : MeasurableSpace α\nμ : Measure α\ninst✝ : Zero β\nhm : m ≤ m₀\ns : Set α\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)\nhf : AEStronglyMeasurable f μ\nhf_zero : f =ᵐ[μ.res... | suffices StronglyMeasurable[m'] (s.indicator (hf.mk f)) from
this.aestronglyMeasurable.congr h_ind_eq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 64
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nν : Measure α\nι : Type u_2\ninst✝ : Countable ι\nμ : ι → Measure α\ns : ι → Set α\nhsm : ∀ (i : ι), MeasurableSet (s i)\nhsμ : ∀ (i : ι), (μ i) (s i) = 0\nhsν : ∀ (i : ι), ν (s i)ᶜ = 0\n⊢ ∀ (i : ι), (μ i) (⋂ b, s b) = 0",
"ppTerm": "?refine_1",
... | [] | exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 283,
"column": 6
} | {
"line": 289,
"column": 15
} | {
"line": 290,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.restrict t)\nh... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.restrict t)\nhS_meas : ∀ (... | obtain ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → x ∈ S m ∩ t := by
rsuffices ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → x ∈ S m
· exact ⟨n, fun m hnm => Set.mem_inter (hn m hnm) hxt⟩
rsuffices ⟨n, hn⟩ : ∃ n, x ∈ S n
· exact ⟨n, fun m hnm => monotone_spanningSets (μ.restrict t) hnm hn⟩
rw [← Set.mem_iUnion, i... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 82
} | {
"line": 234,
"column": 2
} | [
{
"pp": "case hd\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : μ ⟂ₘ ν\nh_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0\nξ : Measure α\nhξμ : ξ ≤ μ\nhξν : ξ ≤ ν\ns : Set α\nhs : MeasurableSet s\n⊢ Disjoint (s ∩ h.nullSet) (s ∩ h.nullSetᶜ)",
"ppTerm": "?hd",
"assigned": true,
"usedConstants... | [
"case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : μ ⟂ₘ ν\nh_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0\nξ : Measure α\nhξμ : ξ ≤ μ\nhξν : ξ ≤ ν\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet (s ∩ h.nullSetᶜ)"
] | · exact Disjoint.mono inter_subset_right inter_subset_right disjoint_compl_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UnitInterval | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 33
} | {
"line": 205,
"column": 34
} | [
{
"pp": "case mp\na t : ℝ\nha : 0 < a\n⊢ a * t ∈ I → t ∈ Icc 0 (1 / a)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Real.instDivInvMonoid",
"Preorder.toLE",
"Membership.mem",
"HDiv.hDiv",
... | [
"case mp\na t : ℝ\nha : 0 < a\nh₁ : 0 ≤ a * t\nh₂ : a * t ≤ 1\n⊢ t ∈ Icc 0 (1 / a)"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.UnitInterval | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 33
} | {
"line": 205,
"column": 34
} | [
{
"pp": "case mpr\na t : ℝ\nha : 0 < a\n⊢ t ∈ Icc 0 (1 / a) → a * t ∈ I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Real.instDivInvMonoid",
"Preorder.toLE",
"Membership.mem",
"HDiv.hDiv",
... | [
"case mpr\na t : ℝ\nha : 0 < a\nh₁ : 0 ≤ t\nh₂ : t ≤ 1 / a\n⊢ a * t ∈ I"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.UnitInterval | {
"line": 212,
"column": 18
} | {
"line": 212,
"column": 33
} | {
"line": 212,
"column": 34
} | [
{
"pp": "case mp\nt : ℝ\n⊢ 2 * t - 1 ∈ I → t ∈ Icc (1 / 2) 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
... | [
"case mp\nt : ℝ\nh₁ : 0 ≤ 2 * t - 1\nh₂ : 2 * t - 1 ≤ 1\n⊢ t ∈ Icc (1 / 2) 1"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.UnitInterval | {
"line": 212,
"column": 18
} | {
"line": 212,
"column": 33
} | {
"line": 212,
"column": 34
} | [
{
"pp": "case mpr\nt : ℝ\n⊢ t ∈ Icc (1 / 2) 1 → 2 * t - 1 ∈ I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Preorder.toLE"... | [
"case mpr\nt : ℝ\nh₁ : 1 / 2 ≤ t\nh₂ : t ≤ 1\n⊢ 2 * t - 1 ∈ I"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.UnitInterval | {
"line": 393,
"column": 4
} | {
"line": 395,
"column": 10
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case neg\na b : ℝ\nx y z : ↑(Icc a b)\ns t : ↑unitInterval\nhs : ↑s = 1\nht : ¬↑t = 1\n⊢ (1 - 1) * ↑x + 1 * ((1 - ↑t) * ↑y + ↑t * ↑z) =\n (1 - 1 * ↑t) * ((1 - 1 * (1 - ↑t) / (1 - 1 * ↑t)) * ↑x + 1 * (1 - ↑t) / (1 - 1 * ↑t) * ↑y) + 1 * ↑t * ↑z",
"ppTerm": "?neg✝",
"assigned": true,
"usedC... | [] | · have : (1 - t : ℝ) ≠ 0 := by grind
field_simp
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UnitInterval | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 28
} | {
"line": 456,
"column": 2
} | [
{
"pp": "a b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\n⊢ ↑y = ↑(convexComb x z ⟨(↑y - ↑x) / (↑z - ↑x), ⋯⟩)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real",
"Set.Icc.convexComb",
"instHDiv",
"Real.instZero",
"Real.instDivInvMonoid",
"Real.... | [
"a b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\n⊢ ↑y = (1 - (↑y - ↑x) / (↑z - ↑x)) * ↑x + (↑y - ↑x) / (↑z - ↑x) * ↑z"
] | simp only [coe_convexComb] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UnitInterval | {
"line": 495,
"column": 2
} | {
"line": 496,
"column": 68
} | {
"line": 497,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nc : ι → Set (↑I × ↑I)\nhc₁ : ∀ (i : ι), IsOpen (c i)\nhc₂ : univ ⊆ ⋃ i, c i\nδ : ℝ\nδ_pos : δ > 0\nball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i\nhδ : 0 < δ / 2\nh : 0 ≤ 1\n⊢ ∃ t,\n ↑(t 0) = ↑0 ∧\n Monotone t ∧ (∃ n, ∀ m ≥ n, ↑(t m) = ↑1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n... | [
"ι : Sort u_1\nc : ι → Set (↑I × ↑I)\nhc₁ : ∀ (i : ι), IsOpen (c i)\nhc₂ : univ ⊆ ⋃ i, c i\nδ : ℝ\nδ_pos : δ > 0\nball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i\nhδ : 0 < δ / 2\nh : 0 ≤ 1\nn m : ℕ\n⊢ ∃ i,\n Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (addNSMul h (δ... | refine ⟨addNSMul h (δ/2), addNSMul_zero h,
monotone_addNSMul h hδ.le, addNSMul_eq_right h hδ, fun n m ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.UnitInterval | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 49
} | {
"line": 547,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : IsTopologicalRing 𝕜\na b : 𝕜\nh : a < b\ne : ↑(Icc 0 1) ≃ₜ ↑(⇑(affineHomeomorph (b - a) a ⋯) '' Icc 0 1) := ⋯\n⊢ ⇑(affineHomeomorph (b - a) a ⋯) '' Icc 0 1 = Icc a b",
... | [
"𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : IsTopologicalRing 𝕜\na b : 𝕜\nh : a < b\ne : ↑(Icc 0 1) ≃ₜ ↑(⇑(affineHomeomorph (b - a) a ⋯) '' Icc 0 1) := (affineHomeomorph (b - a) a ⋯).image (Icc 0 1)\n⊢ Icc a (b - a + a) = Icc a... | rw [affineHomeomorph_image_I _ _ (sub_pos.2 h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 365,
"column": 15
} | {
"line": 365,
"column": 48
} | {
"line": 365,
"column": 48
} | [
{
"pp": "α : Type u_4\nmα : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\ns : Finset α\nhμ : μ = ∑ a ∈ s, μ {a} • dirac a\n⊢ ∀ᵐ (a : α) ∂μ, a ∈ s",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHSMul",
"Meas... | [
"α : Type u_4\nmα : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\ns : Finset α\nhμ : μ = ∑ a ∈ s, μ {a} • dirac a\n⊢ ∀ i ∈ s, ∀ᵐ (x : α) ∂μ {i} • dirac i, x ∈ s"
] | rw [hμ, ae_finsetSum_measure_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 374,
"column": 42
} | {
"line": 374,
"column": 59
} | {
"line": 374,
"column": 59
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ (∀ᵐ (y : α) ∂map f μ, y ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a",
"ppTerm": "?m.60",
"assigned": true,
"... | [
"α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ map f μ = ∑ a ∈ s, (map f μ) {a} • dirac a ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a"
] | ae_mem_finset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Perfect | {
"line": 171,
"column": 2
} | {
"line": 175,
"column": 24
} | {
"line": 176,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : T25Space α\nhC : Perfect C\ny : α\nyC : y ∈ C\n⊢ ∃ C₀ C₁, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Filter.instMemb... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : T25Space α\nhC : Perfect C\ny : α\nyC : y ∈ C\nx : α\nxC : x ∈ C\nhxy : x ≠ y\n⊢ ∃ C₀ C₁, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁"
] | obtain ⟨x, xC, hxy⟩ : ∃ x ∈ C, x ≠ y := by
have := hC.acc _ yC
rw [accPt_iff_nhds] at this
rcases this univ univ_mem with ⟨x, xC, hxy⟩
exact ⟨x, xC.2, hxy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 11
} | {
"line": 223,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\n⊢ ¬SigmaFinite (μ.restrict s)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MeasureTheory.SigmaFini... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\nhsσ : SigmaFinite (μ.restrict s)\n⊢ False"
] | intro hsσ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 11
} | {
"line": 223,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\n⊢ ¬SigmaFinite (μ.restrict s)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MeasureTheory.SigmaFini... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\nhsσ : SigmaFinite (μ.restrict s)\n⊢ False"
] | intro hsσ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 84
} | {
"line": 325,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\n⊢ μ (s ∩ μ.sigmaFiniteSetᶜ) = 0 ∨ μ (s ∩ μ.sigmaFiniteSetᶜ) = ∞",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Compl.compl",
"MeasureTheory.measure_eq_zero_or_top_of_subset_compl_sigma... | [] | exact measure_eq_zero_or_top_of_subset_compl_sigmaFiniteSet Set.inter_subset_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.MetricSpace.Lipschitz | {
"line": 37,
"column": 25
} | {
"line": 46,
"column": 26
} | {
"line": 48,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\ns : Set α\nf : α → β\nhf : LipschitzOnWith K f s\nu : ℕ → α\nhu : CauchySeq u\nh'u : range u ⊆ s\n⊢ CauchySeq (f ∘ u)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.mem_rang... | [] | by
rcases cauchySeq_iff_le_tendsto_0.1 hu with ⟨b, b_nonneg, hb, blim⟩
refine cauchySeq_iff_le_tendsto_0.2 ⟨fun n ↦ K * b n, ?_, ?_, ?_⟩
· exact fun n ↦ mul_nonneg (by positivity) (b_nonneg n)
· intro n m N hn hm
have A n : u n ∈ s := h'u (mem_range_self _)
apply (hf.dist_le_mul _ (A n) _ (A m)).trans
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 337,
"column": 6
} | {
"line": 339,
"column": 64
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∀ {L : ℝ≥0∞},\n L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α),... | [] | apply hL.trans ((ENNReal.add_lt_add hg₁ hg₂).trans_le _)
rw [← lintegral_add_left g₁.measurable.coe_nnreal_ennreal]
simp only [coe_add, Pi.add_apply, ENNReal.coe_add, le_rfl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 337,
"column": 6
} | {
"line": 339,
"column": 64
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∀ {L : ℝ≥0∞},\n L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α),... | [] | apply hL.trans ((ENNReal.add_lt_add hg₁ hg₂).trans_le _)
rw [← lintegral_add_left g₁.measurable.coe_nnreal_ennreal]
simp only [coe_add, Pi.add_apply, ENNReal.coe_add, le_rfl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 59
} | {
"line": 113,
"column": 2
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : Countable ι\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), IsCompletelyPseudoMetrizableSpace (X i)\n⊢ IsCompletelyPseudoMetrizableSpace ((i : ι) → X i)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants"... | [
"X✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : Countable ι\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), IsCompletelyPseudoMetrizableSpace (X i)\nthis : (i : ι) → UpgradedIsCompletelyPseudoMetrizableSpace (X i) := fun i ↦ upgradeIsCompletelyPseudoMetrizable (X i)\n⊢ IsComple... | letI := fun i ↦ upgradeIsCompletelyPseudoMetrizable (X i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 33
} | {
"line": 51,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nC : Set α\nε : ℝ≥0∞\nhC : Perfect C\nε_pos : 0 < ε\nx : α\nxC : x ∈ C\nthis✝ : x ∈ eball x (ε / 2)\nthis : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty\n⊢ let D := closure (eball x (ε / 2) ∩ C);\n Perfect D ∧ D.Nonempty ∧ D ⊆ C... | [
"case refine_1\nα : Type u_1\ninst✝ : MetricSpace α\nC : Set α\nε : ℝ≥0∞\nhC : Perfect C\nε_pos : 0 < ε\nx : α\nxC : x ∈ C\nthis✝ : x ∈ eball x (ε / 2)\nthis : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty\n⊢ closure (eball x (ε / 2) ∩ C) ⊆ C",
"case refine_2\nα : Type u_1\nin... | refine ⟨this.1, this.2, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 272,
"column": 27
} | {
"line": 274,
"column": 51
} | {
"line": 275,
"column": 4
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\np : X\nq : Y\nx✝ : X ⊕ Y\n⊢ Sum.dist (Sum.inl p) x✝ ≤ Sum.dist (Sum.inl p) (Sum.inr q) + Sum.dist (Sum.inr q) x✝",
"ppTerm": "?m.190",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid"... | [] | by
simp only [Sum.dist_eq_glueDist p q]
exact glueDist_triangle _ _ _ (by simp) _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 26
} | {
"line": 337,
"column": 4
} | [
{
"pp": "case mpr.inr\nE : ℕ → Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nf : ((n : ℕ) → E n) → α\nH : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n\nx y : (n : ℕ) → E n\nhne : x ≠ y\n⊢ dist (f x) (f y) ≤ dist x y",
"ppTerm": "?mpr.inr",
"assigned": true,
... | [
"case mpr.inr\nE : ℕ → Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nf : ((n : ℕ) → E n) → α\nH : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n\nx y : (n : ℕ) → E n\nhne : x ≠ y\n⊢ dist (f x) (f y) ≤ (1 / 2) ^ firstDiff x y"
] | rw [dist_eq_of_ne hne] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} → Perf... | [
"case refine_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} → P... | refine ⟨fun x => (inducedMap D).2 ⟨x, hdom⟩, ?_, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 615,
"column": 4
} | {
"line": 616,
"column": 19
} | {
"line": 618,
"column": 0
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nn : ℕ\nx y : X n\n⊢ inductiveLimitDist f ⟨n, x⟩ ⟨n, y⟩ = dist x y",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Real",
... | [] | rw [inductiveLimitDist_eq_dist I ⟨n, x⟩ ⟨n, y⟩ n (le_refl n) (le_refl n), leRecOn_self,
leRecOn_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 92,
"column": 19
} | {
"line": 92,
"column": 57
} | {
"line": 92,
"column": 58
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nF : Set X\nhF : IsClosed[inst✝¹] F\nh : μ Fᶜ = 0\n⊢ F = univ",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"IsClosed.isOpen_compl",
... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nF : Set X\nhF : IsClosed[inst✝¹] F\nh : Fᶜ = ∅\n⊢ F = univ"
] | hF.isOpen_compl.measure_eq_zero_iff μ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Doubling | {
"line": 88,
"column": 35
} | {
"line": 88,
"column": 38
} | {
"line": 88,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝² : PseudoMetricSpace α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsUnifLocDoublingMeasure μ\nK : ℝ\nC : ℝ≥0 := doublingConstant μ\nn : ℕ\nihn : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))\nε : ℝ\nhεn : ∀ (x : α),... | [
"α : Type u_1\ninst✝² : PseudoMetricSpace α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsUnifLocDoublingMeasure μ\nK : ℝ\nC : ℝ≥0 := doublingConstant μ\nn : ℕ\nihn : ∀ᶠ (ε : ℝ) in 𝓝[>] 0, ∀ (x : α), μ (closedBall x (2 ^ n * (2 * ε))) ≤ ↑C ^ n * μ (closedBall x (2 * ε))\nε : ℝ\nhεn : ∀ (x : α), μ (closedBa... | hε, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 38
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurabl... | simp only [← indicator_comp_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 330,
"column": 87
} | {
"line": 345,
"column": 40
} | {
"line": 347,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nt : TopologicalSpace α\ninst✝⁵ : PolishSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\ntβ : TopologicalSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : OpensMeasurableSpace β\nf : α → β\ninst✝ : SecondCountableTopology ↑(range f)\nhf : Measurable f\n⊢ ∃ t' ≤ t, Cont... | [] | by
obtain ⟨b, b_count, -, hb⟩ :
∃ b : Set (Set (range f)), b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b :=
exists_countable_basis (range f)
haveI : Countable b := b_count.to_subtype
have : ∀ s : b, IsClopenable (rangeFactorization f ⁻¹' s) := fun s ↦ by
apply MeasurableSet.isClopenable
exact hf.su... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 40
} | {
"line": 295,
"column": 2
} | [
{
"pp": "X : Type u_4\nY : Type u_5\nm : MeasurableSpace X\nμ : Measure X\nm' : MeasurableSpace Y\nν : Measure Y\nl : Filter X\nl' : Filter Y\ns : Set X\nhs : s ∈ l\nhμs : μ s < ∞\nt : Set Y\nht : t ∈ l'\nhνt : ν t < ∞\n⊢ (μ.prod ν).FiniteAtFilter (l ×ˢ l')",
"ppTerm": "?m.53",
"assigned": true,
"us... | [
"case right\nX : Type u_4\nY : Type u_5\nm : MeasurableSpace X\nμ : Measure X\nm' : MeasurableSpace Y\nν : Measure Y\nl : Filter X\nl' : Filter Y\ns : Set X\nhs : s ∈ l\nhμs : μ s < ∞\nt : Set Y\nht : t ∈ l'\nhνt : ν t < ∞\n⊢ (μ.prod ν) (s ×ˢ t) < ∞"
] | use s ×ˢ t, Filter.prod_mem_prod hs ht | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 566,
"column": 2
} | {
"line": 569,
"column": 23
} | {
"line": 570,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\nC : Set (Set α)\nD : Set (Set β)\nhC : generateFrom C = inst✝¹\nhD : generateFrom D = inst✝\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : μ.FiniteSpanningSetsIn C\nh3D : ν.FiniteSpanningSet... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\nC : Set (Set α)\nD : Set (Set β)\nhC : generateFrom C = inst✝¹\nhD : generateFrom D = inst✝\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : μ.FiniteSpanningSetsIn C\nh3D : ν.FiniteSpanningSetsIn D\nμν : ... | refine
(h3C.prod h3D).ext
(generateFrom_eq_prod hC hD h3C.isCountablySpanning h3D.isCountablySpanning).symm
(h2C.prod h2D) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 701,
"column": 35
} | {
"line": 701,
"column": 88
} | {
"line": 702,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\np : α → β → Prop\nh : MeasurableSet {x | p x.1 x.2}\n⊢ (∀ᵐ (x : α × β) ∂μ.prod ν, p x.1 x.2) ↔ ∀ᵐ (x : β × α) ∂ν.prod μ, p x.2 x.1",
"ppTerm": "?m... | [] | by rw [← prod_swap, ae_map_iff (by fun_prop) h]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Group.Measure | {
"line": 706,
"column": 19
} | {
"line": 706,
"column": 46
} | {
"line": 707,
"column": 2
} | [
{
"pp": "case isOpen\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : TopologicalSpace G\ninst✝² : BorelSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\ns U : Set G\nhU : IsOpen[inst✝³] U\n⊢ U * closure[inst✝³] {1} = U",
"ppTerm": "?isOpen",
"assigned": true,
"usedConstants": [
"IsO... | [] | exact hU.mul_closure_one_eq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Group.Measure | {
"line": 706,
"column": 19
} | {
"line": 706,
"column": 46
} | {
"line": 707,
"column": 2
} | [
{
"pp": "case isOpen\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : TopologicalSpace G\ninst✝² : BorelSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\ns U : Set G\nhU : IsOpen[inst✝³] U\n⊢ U * closure[inst✝³] {1} = U",
"ppTerm": "?isOpen",
"assigned": true,
"usedConstants": [
"IsO... | [] | exact hU.mul_closure_one_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 706,
"column": 19
} | {
"line": 706,
"column": 46
} | {
"line": 707,
"column": 2
} | [
{
"pp": "case isOpen\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : TopologicalSpace G\ninst✝² : BorelSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\ns U : Set G\nhU : IsOpen[inst✝³] U\n⊢ U * closure[inst✝³] {1} = U",
"ppTerm": "?isOpen",
"assigned": true,
"usedConstants": [
"IsO... | [] | exact hU.mul_closure_one_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 219,
"column": 67
} | {
"line": 222,
"column": 25
} | {
"line": 224,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\nf : α → ℝ≥0∞\n⊢ (μ.withDensity f).restrict s = (μ.restrict s).withDensity f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureT... | [] | by
ext1 t ht
rw [restrict_apply ht, withDensity_apply _ ht, withDensity_apply' _ (t ∩ s),
restrict_restrict ht] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 40
} | {
"line": 887,
"column": 2
} | [
{
"pp": "γ : Type u_3\nt t' : TopologicalSpace γ\nht : PolishSpace γ\nht' : PolishSpace γ\nhle : t ≤ t'\n⊢ borel γ = borel γ",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MeasurableSpace.instPartialOrder",
"borel",
"MeasurableSpace",
"le_antisymm",
"borel_an... | [
"γ : Type u_3\nt t' : TopologicalSpace γ\nht : PolishSpace γ\nht' : PolishSpace γ\nhle : t ≤ t'\n⊢ borel γ ≤ borel γ"
] | refine le_antisymm ?_ (borel_anti hle) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Complex.Basic | {
"line": 849,
"column": 67
} | {
"line": 860,
"column": 30
} | {
"line": 862,
"column": 0
} | [
{
"pp": "a b₁ b₂ : ℝ\n⊢ (fun y ↦ ↑a + ↑y * I) '' [[b₁, b₂]] = {a} ×ℂ [[b₁, b₂]]",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"Real",
"Complex.mul_re",
"Complex.equivRealProd_apply",
"... | [] | by
rw [← preimage_equivRealProd_prod]
ext x
constructor
· intro hx
obtain ⟨x₁, hx₁, hx₁'⟩ := hx
simp [← hx₁', mem_preimage, mem_prod, hx₁]
· intro hx
simp only [equivRealProd_apply, singleton_prod, mem_image, Prod.mk.injEq,
exists_eq_right_right, mem_preimage] at hx
obtain ⟨x₁, hx₁, hx₁'... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Order | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 18
} | [
{
"pp": "z : ℂ\n⊢ z ^ 2 ≤ 0 ↔ z.re = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Complex.im",
"PartialOrder.toPreorder",
"Complex.instZero",
"Preorder.toLE",
"id... | [
"z : ℂ\n⊢ (z ^ 2).re ≤ 0 ∧ (z ^ 2).im = 0 ↔ z.re = 0"
] | nonpos_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Norm | {
"line": 382,
"column": 35
} | {
"line": 382,
"column": 62
} | {
"line": 382,
"column": 62
} | [
{
"pp": "x : ℝ\nhx : ‖x‖ ≤ 1\n⊢ 0 ≤ 1 - x ^ 2",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toA... | [] | by nlinarith [abs_le.mp hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Order | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 33
} | {
"line": 102,
"column": 0
} | [
{
"pp": "z w : ℂ\n⊢ ¬z ≤ w ↔ w.re < z.re ∨ z.im ≠ w.im",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Rea... | [] | rw [le_def, not_and_or, not_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Order | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 33
} | {
"line": 102,
"column": 0
} | [
{
"pp": "z w : ℂ\n⊢ ¬z ≤ w ↔ w.re < z.re ∨ z.im ≠ w.im",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Rea... | [] | rw [le_def, not_and_or, not_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Order | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 33
} | {
"line": 102,
"column": 0
} | [
{
"pp": "z w : ℂ\n⊢ ¬z ≤ w ↔ w.re < z.re ∨ z.im ≠ w.im",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"not_le",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Rea... | [] | rw [le_def, not_and_or, not_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 760,
"column": 31
} | {
"line": 760,
"column": 47
} | {
"line": 760,
"column": 47
} | [
{
"pp": "M : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\nμ ν : Measure M\ninst✝ : SFinite ν\ns : ℝ≥0∞\n⊢ map (fun x ↦ x.1 * x.2) (s • μ.prod ν) = s • map (fun x ↦ x.1 * x.2) (μ.prod ν)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Me... | [
"M : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\nμ ν : Measure M\ninst✝ : SFinite ν\ns : ℝ≥0∞\n⊢ s • map (fun x ↦ x.1 * x.2) (μ.prod ν) = s • map (fun x ↦ x.1 * x.2) (μ.prod ν)"
] | Measure.map_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 404,
"column": 8
} | {
"line": 404,
"column": 29
} | {
"line": 404,
"column": 30
} | [
{
"pp": "V : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g✝ f g : NormedAddGroupHom V₁ V₂\n⊢ ∃ C, ∀ (v : V₁), ‖(f.toAddMonoidHom + -g.toAddMonoidHom) v‖ ≤ C... | [] | exact (f + -g).bound' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Algebra.Equiv | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 30
} | {
"line": 89,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : Semiring B\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring C\ninst✝³ : TopologicalSpace C\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R C\ng' : A ... | [
"R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : Semiring B\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : Semiring C\ninst✝³ : TopologicalSpace C\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R C\ntoFun✝¹ : A → B\nin... | rcases g' with ⟨⟨_, _⟩, _⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.Basic | {
"line": 74,
"column": 33
} | {
"line": 74,
"column": 43
} | {
"line": 74,
"column": 44
} | [
{
"pp": "z : ℂ\nR : Type u_1\ninst✝¹ : NormedField R\ninst✝ : NormedAlgebra R ℝ\nr : R\nx : ℂ\n⊢ ‖(algebraMap R ℝ) r • x‖ ≤ ‖r‖ * ‖x‖",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.real_smul",
... | [
"z : ℂ\nR : Type u_1\ninst✝¹ : NormedField R\ninst✝ : NormedAlgebra R ℝ\nr : R\nx : ℂ\n⊢ ‖↑((algebraMap R ℝ) r) * x‖ ≤ ‖r‖ * ‖x‖"
] | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Basic | {
"line": 558,
"column": 2
} | {
"line": 558,
"column": 17
} | {
"line": 559,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → 𝕜\nc : 𝕜\n⊢ HasSum (fun x ↦ re (f x)) (re c) L ∧ HasSum (fun x ↦ im (f x)) (im c) L → HasSum f c L",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"R... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → 𝕜\nc : 𝕜\nh₁ : HasSum (fun x ↦ re (f x)) (re c) L\nh₂ : HasSum (fun x ↦ im (f x)) (im c) L\n⊢ HasSum f c L"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.RCLike.Basic | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 40
} | {
"line": 237,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nr : ℝ\n⊢ im (z * ↑r) = im z * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | [] | rw [mul_comm, im_ofReal_mul, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 40
} | {
"line": 237,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nr : ℝ\n⊢ im (z * ↑r) = im z * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | [] | rw [mul_comm, im_ofReal_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 40
} | {
"line": 237,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nr : ℝ\n⊢ im (z * ↑r) = im z * r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | [] | rw [mul_comm, im_ofReal_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 27
} | {
"line": 239,
"column": 28
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nz : K\n⊢ re (r • z) = r * re z",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"AddMonoid.t... | [
"K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nz : K\n⊢ re (↑r * z) = r * re z"
] | real_smul_eq_coe_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 27
} | {
"line": 243,
"column": 28
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nz : K\n⊢ im (r • z) = r * im z",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"RCLike.toNormedAlgebra",
"HMul.hMul",
"AddMonoid.t... | [
"K : Type u_1\ninst✝ : RCLike K\nr : ℝ\nz : K\n⊢ im (↑r * z) = r * im z"
] | real_smul_eq_coe_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 505,
"column": 2
} | {
"line": 506,
"column": 17
} | {
"line": 509,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ im (z / w) = im z * re w / normSq w - re z * im w / normSq w",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssoc... | [] | simp only [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, neg_mul, mul_neg,
rclike_simps] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 505,
"column": 2
} | {
"line": 506,
"column": 17
} | {
"line": 509,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ im (z / w) = im z * re w / normSq w - re z * im w / normSq w",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssoc... | [] | simp only [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, neg_mul, mul_neg,
rclike_simps] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 505,
"column": 2
} | {
"line": 506,
"column": 17
} | {
"line": 509,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ im (z / w) = im z * re w / normSq w - re z * im w / normSq w",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssoc... | [] | simp only [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, neg_mul, mul_neg,
rclike_simps] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 808,
"column": 69
} | {
"line": 808,
"column": 88
} | {
"line": 810,
"column": 0
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝³ : RCLike K\ninst✝² : NormedField E\ninst✝¹ : CharZero E\ninst✝ : NormedSpace K E\nι : Type u_3\ns : Finset ι\nf : ι → E\nx✝¹ : ℕ\nx✝ : E\n⊢ ‖(↑x✝¹)⁻¹ • x✝‖ = (↑x✝¹)⁻¹ • ‖x✝‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | rw [norm_nnqsmul K] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 808,
"column": 69
} | {
"line": 808,
"column": 88
} | {
"line": 810,
"column": 0
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝³ : RCLike K\ninst✝² : NormedField E\ninst✝¹ : CharZero E\ninst✝ : NormedSpace K E\nι : Type u_3\ns : Finset ι\nf : ι → E\nx✝¹ : ℕ\nx✝ : E\n⊢ ‖(↑x✝¹)⁻¹ • x✝‖ = (↑x✝¹)⁻¹ • ‖x✝‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | rw [norm_nnqsmul K] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 808,
"column": 69
} | {
"line": 808,
"column": 88
} | {
"line": 810,
"column": 0
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝³ : RCLike K\ninst✝² : NormedField E\ninst✝¹ : CharZero E\ninst✝ : NormedSpace K E\nι : Type u_3\ns : Finset ι\nf : ι → E\nx✝¹ : ℕ\nx✝ : E\n⊢ ‖(↑x✝¹)⁻¹ • x✝‖ = (↑x✝¹)⁻¹ • ‖x✝‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | rw [norm_nnqsmul K] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 903,
"column": 2
} | {
"line": 903,
"column": 34
} | {
"line": 904,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nhz : 0 < z\n⊢ 0 < z⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"RCLike.pos_iff_exists_ofReal",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\nhz : ∃ x > 0, ↑x = z\n⊢ 0 < z⁻¹"
] | rw [pos_iff_exists_ofReal] at hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 70,
"column": 81
} | {
"line": 71,
"column": 86
} | {
"line": 73,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\nhx : x ∈ e.source\n⊢ Tendsto (↑e.symm) (𝓝 (↑e x)) (𝓝 x)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"ContinuousAt",
"nhds"... | [] | by
simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 990,
"column": 4
} | {
"line": 990,
"column": 21
} | {
"line": 991,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\na : { x // 0 < x }\nb c : K\nh : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) a b) ((fun x y ↦ ↑x * y) a c)\na' : ℝ\nha1 : a' > 0\nha2 : ↑a' = ↑a\n⊢ b ≤ c",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGr... | [
"K : Type u_1\ninst✝ : RCLike K\na : { x // 0 < x }\nb c : K\nh : (fun x1 x2 ↦ x1 ≤ x2) ((fun x y ↦ ↑x * y) a b) ((fun x y ↦ ↑x * y) a c)\na' : ℝ\nha1 : a' > 0\nha2 : ↑a' = ↑a\n⊢ 0 ≤ c - b"
] | rw [← sub_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1184,
"column": 21
} | {
"line": 1184,
"column": 46
} | {
"line": 1184,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re (x - y)‖ₑ ≤ ‖x - y‖ₑ",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.i... | [
"K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re (x - y)‖ ≤ ‖x - y‖"
] | rw [enorm_le_iff_norm_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1191,
"column": 21
} | {
"line": 1191,
"column": 46
} | {
"line": 1191,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖im (x - y)‖ₑ ≤ ‖x - y‖ₑ",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.i... | [
"K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖im (x - y)‖ ≤ ‖x - y‖"
] | rw [enorm_le_iff_norm_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1191,
"column": 18
} | {
"line": 1191,
"column": 77
} | {
"line": 1193,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖im (x - y)‖ₑ ≤ ‖x - y‖ₑ",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.i... | [] | by rw [enorm_le_iff_norm_le]; exact norm_im_le_norm (x - y) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 731,
"column": 2
} | {
"line": 731,
"column": 50
} | {
"line": 733,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\ng : α → F\nf' : α → E'\nl : Filter α\n⊢ (∃ c, IsBigOWith c l (fun x ↦ ‖f' x‖) g) ↔ ∃ c, IsBigOWith c l f' g",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real... | [] | exact exists_congr fun _ => isBigOWith_norm_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 96
} | {
"line": 160,
"column": 2
} | [
{
"pp": "α : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf'' : α → E''\nl : Filter α\nc : F''\n⊢ (f'' =O[l] fun _x ↦ c) ↔ (c = 0 → f'' =ᶠ[l] 0) ∧ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖f'' x‖",
"ppTerm": "?m.25",
"assigned": true,
... | [
"α : Type u_1\nE'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf'' : α → E''\nl : Filter α\nc : F''\n⊢ ((c = 0 → f'' =ᶠ[l] 0) ∧ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖f'' x‖) → f'' =O[l] fun _x ↦ c"
] | refine ⟨fun h => ⟨fun hc => isBigO_zero_right_iff.1 (by rwa [← hc]), h.isBoundedUnder_le⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 241,
"column": 61
} | {
"line": 242,
"column": 83
} | {
"line": 244,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ cosh x ^ 2 - sinh x ^ 2 = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"add_neg_cancel",
"Complex.sinh",
"HMul.hMul",
"Complex.commRing",
"Complex.exp_add",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Com... | [] | by
rw [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_cancel, exp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Exponential | {
"line": 360,
"column": 6
} | {
"line": 360,
"column": 64
} | {
"line": 361,
"column": 6
} | [
{
"pp": "case hba\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn j : ℕ\nhn : 0 < n\nx : ℕ\na✝ : x ∈ range (j - n)\n⊢ ↑n.factorial * ↑n.succ ^ x ≤ ↑(x + n).factorial",
"ppTerm": "?hba",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemi... | [
"case hba\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn j : ℕ\nhn : 0 < n\nx : ℕ\na✝ : x ∈ range (j - n)\n⊢ n.factorial * n.succ ^ x ≤ (n + x).factorial"
] | rw [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 524,
"column": 2
} | {
"line": 530,
"column": 31
} | {
"line": 532,
"column": 0
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ rexp x ≤ ∑ m ∈ range n, x ^ m / ↑m.factorial + x ^ n * (↑n + 1) / (↑n.factorial * ↑n)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"... | [] | have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
have h'' := (abs_sub_le_iff.1 h').1
have t := sub_le_iff_le_add'.1 h''
simpa [mul_div_assoc] using t | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Exponential | {
"line": 524,
"column": 2
} | {
"line": 530,
"column": 31
} | {
"line": 532,
"column": 0
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ rexp x ≤ ∑ m ∈ range n, x ^ m / ↑m.factorial + x ^ n * (↑n + 1) / (↑n.factorial * ↑n)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"... | [] | have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
have h'' := (abs_sub_le_iff.1 h').1
have t := sub_le_iff_le_add'.1 h''
simpa [mul_div_assoc] using t | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 99
} | {
"line": 122,
"column": 2
} | [
{
"pp": "f : ℕ → ℝ\nR : ℝ\nA : Set.Ico 0 R ⊆ Set.Ioo (-R) R\nB : Set.Ioo 0 R ⊆ Set.Ioo (-R) R\ntfae_1_to_3 : (∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =O[atTop] fun x ↦ a ^ x\ntfae_2_to_1 : (∃ a ∈ Set.Ioo 0 R, f =o[atTop] fun x ↦ a ^ x) → ∃ a ∈ Set.Ioo (-R) R, f =o[atTop] fun x... | [] | simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne') | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 307,
"column": 10
} | {
"line": 307,
"column": 63
} | {
"line": 307,
"column": 63
} | [
{
"pp": "α : Type u_1\nx y z : ℝ\nl : Filter α\n⊢ Tendsto (fun x ↦ ⟨rexp x, ⋯⟩) atTop atTop",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | [] | rw [tendsto_Ioi_atTop]; simp only [tendsto_exp_atTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 307,
"column": 10
} | {
"line": 307,
"column": 63
} | {
"line": 307,
"column": 63
} | [
{
"pp": "α : Type u_1\nx y z : ℝ\nl : Filter α\n⊢ Tendsto (fun x ↦ ⟨rexp x, ⋯⟩) atTop atTop",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | [] | rw [tendsto_Ioi_atTop]; simp only [tendsto_exp_atTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 280,
"column": 2
} | {
"line": 283,
"column": 68
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁷ : Field R✝\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\nv w : AbsoluteValue R✝ S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\n_i : OrderTopology S\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\n⊢ Ha... | [] | constructor
intro x hx
have h1 : Summable fun n : ℕ ↦ ‖x‖ ^ n := summable_geometric_of_lt_one (norm_nonneg _) hx
exact h1.of_norm_bounded_eventually_nat (eventually_norm_pow_le x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 280,
"column": 2
} | {
"line": 283,
"column": 68
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁷ : Field R✝\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\nv w : AbsoluteValue R✝ S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\n_i : OrderTopology S\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\n⊢ Ha... | [] | constructor
intro x hx
have h1 : Summable fun n : ℕ ↦ ‖x‖ ^ n := summable_geometric_of_lt_one (norm_nonneg _) hx
exact h1.of_norm_bounded_eventually_nat (eventually_norm_pow_le x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 221,
"column": 47
} | {
"line": 221,
"column": 94
} | {
"line": 223,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s / closedBall 1 δ = cthickening δ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"IsCompact.mul_closedBall_one",
... | [] | simp [div_eq_mul_inv, hs.mul_closedBall_one hδ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 221,
"column": 47
} | {
"line": 221,
"column": 94
} | {
"line": 223,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s / closedBall 1 δ = cthickening δ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"IsCompact.mul_closedBall_one",
... | [] | simp [div_eq_mul_inv, hs.mul_closedBall_one hδ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 221,
"column": 47
} | {
"line": 221,
"column": 94
} | {
"line": 223,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s / closedBall 1 δ = cthickening δ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"IsCompact.mul_closedBall_one",
... | [] | simp [div_eq_mul_inv, hs.mul_closedBall_one hδ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.PathConnected | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 14
} | {
"line": 474,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx : X\n⊢ ∀ ⦃y : X⦄, y ∈ {x} → JoinedIn {x} x y",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"JoinedIn",
"Membership.mem",
"Set.instSingletonSet",
"Eq.ndrec",
"Singleton.singleton",
"Set.instMembership... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ny : X\n⊢ JoinedIn {y} y y"
] | rintro y rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Connected.PathConnected | {
"line": 471,
"column": 77
} | {
"line": 474,
"column": 25
} | {
"line": 476,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx : X\n⊢ IsPathConnected {x}",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"JoinedIn",
"Membership.mem",
"Set.instSingletonSet",
"And",
"And.intro",
"Exists.intro",
"Eq.ndrec",
"Singleton.si... | [] | by
refine ⟨x, rfl, ?_⟩
rintro y rfl
exact JoinedIn.refl rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 401,
"column": 51
} | {
"line": 401,
"column": 71
} | {
"line": 401,
"column": 71
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 ≤ r\nx : E\n⊢ x +ᵥ closedBall 0 r = closedBall x r",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instVAddOfAdd",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 ≤ r\nx : E\n⊢ closedBall x r = closedBall x r"
] | vadd_closedBall_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 79,
"column": 2
} | {
"line": 83,
"column": 69
} | {
"line": 84,
"column": 2
} | [
{
"pp": "case refine_1\nx r : ℝ\nhr : r ≤ ‖↑x‖\n⊢ r ≤ |x - ↑(round x)|",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",... | [
"case refine_2\nx : ℝ\n⊢ ∀ (m : ℝ), ↑m = ↑x → |x - ↑(round x)| ≤ |m|"
] | · rw [abs_sub_round_eq_min, le_inf_iff]
rw [le_norm_iff] at hr
constructor
· simpa [abs_of_nonneg] using hr (fract x)
· simpa [abs_sub_comm (fract x)] using hr (fract x - 1) (by simp) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Path | {
"line": 582,
"column": 53
} | {
"line": 582,
"column": 69
} | {
"line": 582,
"column": 69
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type u_3\nγ✝ : Path x y\na b : X\nγ : Path a b\nt : ℝ\n⊢ γ.extend (min t t) = γ.extend t",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrAr... | [] | by rw [min_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 132,
"column": 34
} | {
"line": 133,
"column": 87
} | {
"line": 135,
"column": 0
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\nx✝ : AddCircle p\nε : ℝ\nhε : |p| / 2 ≤ ε\nx : AddCircle p\n⊢ x ∈ closedBall x✝ ε",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"instHDiv",... | [] | by
simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 22
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ 0 < log 0 ↔ 1 < 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
"Preor... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 22
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ 0 < log 0 ↔ 1 < 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
"Preor... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 22
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ 0 < log 0 ↔ 1 < 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
"Preor... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 22
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ log 0 ≤ 0 ↔ 0 ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"instReflLe",
"congrArg",
"Std.le_refl._simp_1",
"zero_le_one._simp_1",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 22
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ log 0 ≤ 0 ↔ 0 ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"instReflLe",
"congrArg",
"Std.le_refl._simp_1",
"zero_le_one._simp_1",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 22
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case inl\nhx : 0 ≤ 0\n⊢ log 0 ≤ 0 ↔ 0 ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"instReflLe",
"congrArg",
"Std.le_refl._simp_1",
"zero_le_one._simp_1",
... | [] | simp [zero_le_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 120,
"column": 47
} | {
"line": 120,
"column": 65
} | {
"line": 122,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ x ^ 0 = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"Real.instZero",
"congrArg",
"Complex.instPow",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"HPow.hPow",
"True",
"eq_sel... | [] | by simp [rpow_def] | [anonymous] | Lean.Parser.Term.byTactic |
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